Linear Algebra

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$MATLAB C++ Math Library Reference$

328 Chapter 4. Determinants by expanding along the first row, as in Example 1.1. � � |T | =1· (+1) �5 �8 � � 6� � � 9� +2· (−1) �4 �7 � � 6� � � 9� +3· (+1) �4 �7 � 5� � 8� = −3+12−9=0 Alternatively, we can expand down the second column. � � |T | =2· (−1) �4 �7 � � 6� � � 9� +5· (+1) �1 �7 � � 3� � � 9� +8· (−1) �1 �4 � 3� � 6� =12−60 + 48 = 0 1.7 Example A row or column with many zeroes suggests a Laplace expansion. �� 1 2 3 5 1 −1 � 0� � � � 1� � =0· (+1) �2 � 0� 3 � � 1 � � � −1� +1· (−1) �1 �3 � � 5 � � � −1� +0· (+1) �1 �2 � 5� � 1� =16 We finish by applying this result to derive a new formula for the inverse of a matrix. With Theorem 1.5, the determinant of an n × n matrix T can be calculated by taking linear combinations of entries from a row and their associated cofactors. ti,1 · Ti,1 + ti,2 · Ti,2 + ···+ ti,n · Ti,n = |T | (∗) Recall that a matrix with two identical rows has a zero determinant. Thus, for any matrix T , weighing the cofactors by entries from the “wrong” row — row k with k �= i — gives zero ti,1 · Tk,1 + ti,2 · Tk,2 + ···+ ti,n · Tk,n =0 (∗∗) because it represents the expansion along the row k of a matrix with row i equal to row k. This equation summarizes (∗) and (∗∗). ⎛ ⎜ ⎝ t1,1 t2,1 t1,2 t2,2 . ... ... t1,n t2,n ⎞ ⎛ T1,1 ⎟ ⎜T1,2 ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ T2,1 T2,2 . ... ... ⎞ Tn,1 Tn,2 ⎟ ⎠ = ⎛ |T | ⎜ 0 ⎜ ⎝ 0 |T | . ... ... ⎞ 0 0 ⎟ ⎠ 0 0 ... |T | tn,1 tn,2 ... tn,n T1,n T2,n ... Tn,n Note that the order of the subscripts in the matrix of cofactors is opposite to the order of subscripts in the other matrix; e.g., along the first row of the matrix of cofactors the subscripts are 1, 1 then 2, 1, etc. 1.8 Definition The matrix adjoint to the square matrix T is ⎛ T1,1 ⎜T1,2 ⎜ adj(T )= ⎜ ⎝ T2,1 T2,2 . ... ... ⎞ Tn,1 Tn,2 ⎟ ⎠ where Tj,i is the j, i cofactor. T1,n T2,n ... Tn,n
Section III. Other Formulas 329 1.9 Theorem Where T is a square matrix, T · adj(T )=adj(T ) · T = |T |·I. Proof. Equations (∗) and (∗∗). QED 1.10 Example If ⎛ 1 T = ⎝2 0 1 ⎞ 4 −1⎠ 1 0 1 then the adjoint adj(T )is ⎛ T1,1 ⎝T1,2 T1,3 T2,1 T2,2 T2,3 ⎛ � � �1 ⎞ ⎜ � ⎜ 0 T3,1 ⎜ T3,2⎠= ⎜ T3,3 ⎜ ⎝ � −1� � 1 � − � � − �2 �1 � −1� � 1 � � � �0 �0 � � �1 �1 � 4� � 1� � 4� � 1� � � �0 �1 � 4 � � −1� − � � �2 �1 � 1� � 0� � � − �1 �1 � 0� � 0� � � �1 �2 � � �1 �2 ⎞ ⎟ �⎟ ⎛ 4 �⎟ 1 �⎟ −1�⎟= ⎝−3 ⎟ � ⎟ −1 0�⎠ � 1� 0 −3 0 ⎞ −4 9 ⎠ 1 and taking the product with T gives the diagonal matrix |T |·I. ⎛ 1 ⎝2 0 1 ⎞ ⎛ 4 1 −1⎠ ⎝−3 0 −3 ⎞ ⎛ −4 −3 9 ⎠ = ⎝ 0 0 −3 ⎞ 0 0 ⎠ 1 0 1 −1 0 1 0 0 −3 1.11 Corollary If |T |�= 0 then T −1 =(1/|T |) · adj(T ). 1.12 Example The inverse of the matrix from Example 1.10 is (1/−3)·adj(T ). T −1 ⎛ 1/−3 = ⎝−3/−3 0/−3 −3/−3 ⎞ ⎛ −4/−3 −1/3 9/−3⎠ = ⎝ 1 0 1 ⎞ 4/3 −3 ⎠ −1/−3 0/−3 1/−3 1/3 0 −1/3 The formulas from this section are often used for by-hand calculation and are sometimes useful with special types of matrices. However, they are not the best choice for computation with arbitrary matrices because they require more arithmetic than, for instance, the Gauss-Jordan method. Exercises � 1.13 Find the cofactor. T = � 1 0 � 2 −1 1 3 0 2 −1 (a) T2,3 (b) T3,2 (c) T1,3 � 1.14 Find the determinant by expanding � � 3 0 � � 1 2 � −1 3 � 1� � 2� 0 �
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Preface In most mathematics program
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dent projects for individuals or sm
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Contents 1 Linear Systems 1 1.I Sol
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5.II.1 Definition and Examples ....
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2 Chapter 1. Linear Systems To fini
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4 Chapter 1. Linear Systems Conside
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22 Chapter 1. Linear Systems Studyi
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28 Chapter 1. Linear Systems 3.13 E
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30 Chapter 1. Linear Systems (a) 2x
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32 Chapter 1. Linear Systems 1.II L
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34 Chapter 1. Linear Systems positi
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36 Chapter 1. Linear Systems In R3
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38 Chapter 1. Linear Systems � 1.
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40 Chapter 1. Linear Systems Notice
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42 Chapter 1. Linear Systems 2.7 De
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56 Chapter 1. Linear Systems First,
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80 Chapter 2. Vector Spaces 2.I Def
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82 Chapter 2. Vector Spaces For the
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84 Chapter 2. Vector Spaces A vecto
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86 Chapter 2. Vector Spaces 1.12 Ex
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88 Chapter 2. Vector Spaces Chapter
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90 Chapter 2. Vector Spaces (d) The
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92 Chapter 2. Vector Spaces 2.2 Exa
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94 Chapter 2. Vector Spaces second.
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96 Chapter 2. Vector Spaces Proof.
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98 Chapter 2. Vector Spaces The sub
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100 Chapter 2. Vector Spaces S. Mem
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102 Chapter 2. Vector Spaces 2.II L
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104 Chapter 2. Vector Spaces 1.5 Ex
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106 Chapter 2. Vector Spaces Lookin
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108 Chapter 2. Vector Spaces Proof.
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110 Chapter 2. Vector Spaces (a) {2
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112 Chapter 2. Vector Spaces � 1.
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128 Chapter 2. Vector Spaces Anothe
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130 Chapter 2. Vector Spaces (a)
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136 Chapter 2. Vector Spaces 4.12 E
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160 Chapter 3. Maps Between Spaces
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Section IV. Jordan Form 379 5.IV Jo
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Topic: Computing Eigenvalues—the
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Topic: Computing Eigenvalues—the
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Topic: Stable Populations 403 Topic
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Topic: Linear Recurrences 405 Topic
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Topic: Linear Recurrences 407 And,
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Topic: Linear Recurrences 409 diamo
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Topic: Linear Recurrences 411 Compu
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Appendix Introduction Mathematics i
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A-3 ‘if a number is divisible by
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Techniques of Proof A-5 Induction.
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A-7 The Principle of Extensionality
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A-9 So an identity map plays the sa
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A-11 Similarly, the equivalence rel
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Bibliography [Abbot] Edwin Abbott,
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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congruent plane figures, 286 contra
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Gauss’ method, 3 Gauss-Jordan, 46
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Linear Algebra

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